The proper geometric dimension of the mapping class group of an orientable surface with punctures

TL;DR

This paper proves that the full mapping class group of any punctured orientable surface has a cocompact classifying space for proper actions matching its virtual cohomological dimension, extending previous results.

Contribution

It establishes the proper geometric dimension for the full mapping class group of punctured surfaces, generalizing prior work on closed surfaces and pure groups.

Findings

The full mapping class group admits a cocompact classifying space of dimension equal to its virtual cohomological dimension.

The result applies to all orientable surfaces with punctures, not just closed ones.

It determines the proper geometric dimension of full spherical braid groups.

Abstract

We show that the {\it full} mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for {\it pure} mapping class groups by Aramayona and Mart\'inez P\'erez. As a consequence of our result we also obtain the proper geometric dimension of {\it full} spherical braid groups.